(The centrally synergistic potential function \(\mathcal {U}(e,h)\) ):
The potential function \(\mathcal {U}(e,h) = \Vert e\Vert ^{2}\) , for \(e=q-hq_{d}\)  can be expressed in terms of the scalar part \(\varepsilon _0\)  of the quaternion error \(\varepsilon \)  (REF ) as \(\mathcal {U}(e(\varepsilon _0),h) = 2(1-h\varepsilon _{0}) :=\mathcal {U}_\varepsilon (\varepsilon _0,h)\)  (shown in Fig. REF ). This potential function has been widely used to globally stabilize the set \(\mathcal {A} = \lbrace  \pm \hat{1} \rbrace \times \mathcal {H}\) , where \(\hat{1} = [1,0,0,0]^{T}\in \mathcal {S}^{3}\) , see for instance [1]}, [2]}, [3]}. However, the potential function \(\mathcal {U}_\varepsilon (\varepsilon _0,h)\)  relative to \(\mathcal {A}\)  is not central The reader is referred to [1]}, [5]} for more details., i.e., the potential functions \(\varepsilon \mapsto \mathcal {U}_\varepsilon (\varepsilon _0,h)\) , for \(h\in \mathcal {H}\) , have different critical points \(\pm \hat{1}\) . On the other hand, the potential function \(\mathcal {U}(e,h) = \Vert e\Vert ^{2}\)  relative to \(\mathcal {A}_{0}\)  is central, for having the same critical point \(e=0_{4\times 1}\)  for \(h\in \mathcal {H}\) , which guarantees that each mode of the hybrid closed-loop system is almost globally exponentially stable [5]}.
